60.5.4 problem 1537

Internal problem ID [11501]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1537
Date solved : Sunday, March 30, 2025 at 08:23:44 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-12 y^{\prime \prime }+12 y-16 x^{4} {\mathrm e}^{x^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 67

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-12*diff(diff(y(x),x),x)+12*y(x)-16*x^4*exp(x^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x^{2}}+c_1 \,{\mathrm e}^{\sqrt {6-2 \sqrt {6}}\, x}+c_2 \,{\mathrm e}^{\sqrt {6+2 \sqrt {6}}\, x}+c_3 \,{\mathrm e}^{-\sqrt {6-2 \sqrt {6}}\, x}+c_4 \,{\mathrm e}^{-\sqrt {6+2 \sqrt {6}}\, x} \]

✓ Mathematica. Time used: 1.037 (sec). Leaf size: 359

ode=-16*E^x^2*x^4 + 12*y[x] - 12*D[y[x],{x,2}] + Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{\sqrt {6-2 \sqrt {6}} x} \int _1^x-\frac {1}{3} \sqrt {3+\sqrt {6}} e^{-\left (\left (\sqrt {6-2 \sqrt {6}}-K[1]\right ) K[1]\right )} K[1]^4dK[1]+e^{-\sqrt {6-2 \sqrt {6}} x} \int _1^x\frac {1}{3} \sqrt {3+\sqrt {6}} e^{K[2] \left (K[2]+\sqrt {6-2 \sqrt {6}}\right )} K[2]^4dK[2]+e^{\sqrt {2 \left (3+\sqrt {6}\right )} x} \int _1^x\frac {1}{3} \sqrt {3-\sqrt {6}} e^{K[3] \left (K[3]-\sqrt {2 \left (3+\sqrt {6}\right )}\right )} K[3]^4dK[3]+e^{-\sqrt {2 \left (3+\sqrt {6}\right )} x} \int _1^x-\frac {1}{3} \sqrt {3-\sqrt {6}} e^{K[4] \left (K[4]+\sqrt {2 \left (3+\sqrt {6}\right )}\right )} K[4]^4dK[4]+c_1 e^{\sqrt {6-2 \sqrt {6}} x}+c_2 e^{-\sqrt {6-2 \sqrt {6}} x}+c_3 e^{\sqrt {2 \left (3+\sqrt {6}\right )} x}+c_4 e^{-\sqrt {2 \left (3+\sqrt {6}\right )} x} \]

✓ Sympy. Time used: 4.502 (sec). Leaf size: 1168

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-16*x**4*exp(x**2) + 12*y(x) - 12*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]